Wireless subwoofer has blinking blue light and not connecting to soundbar. cannot hear subwoofer

The sound bar and the subwoofer could of had quit working and and no longer works anymore. You will have to by a new soundbar and see if it can work.

I have no experience of wireless speakers but I think you should contact the seller, if the unit is still under warranty or perhaps the support section of the manufacturer's website.
Certainly do read the instrucion manual.

I am guessing but either the wireless signal is being lost (depends on the function of the blue light) or the subwoofer is malfunctioning after a few minutes and the built-in thermal or overload protection is switching it off. The thump sound could be it trying to switch on again and finding the fault is still present.

Just my thoughts...

PAIRING THE JBL SB400 FOR WIRELESS OPERATION
After turning the soundbar and subwoofer on for the first time, you will need to "pair" the subwoofer and soundbar so they both operate at the same wireless frequency. To pair the units, press the Wireless Pairing switches on both units within 30 seconds of each other. The units will communicate with each other and operate at the same frequency.

I presume you need link the Wireless Subwoofer to tour Soundbar.
Here are instructions:
How To Link The Wireless Subwoofer To Your Soundbar HW Models Samsung

turn the soundbar off, then press and hold the reset button on the back of the subwoofer until the blue light starts blinking fast. the point the remote at the soundbar and hold the mute button until"ID SET" shows on the display. Then turn on the soundbar and it should find the sub and pair with it.

Check the receiver in the speaker and the amplifier. they both have to work to get sound.

Push in and hold the switch on the back of the subwoofer until the light starts blinking fast. Then, with the soundbar power turned off, hold the mute button on the remote. "ID SET" should show up on the display. Then turn on the soundbar, and the subwoofer should link to it.

Press and hold the reset button on the back of the subwoofer until the light on the front starts blinking. Then with the soundbar turned on, press the stop button on the soundbar and the mute button on the remote at the same time. The display should say "REMATE". The sub light should stop blinking then and should be linked.

No adapter. Your sound bar has to have wireless built in.